{"id":47386,"date":"2013-10-11T09:28:02","date_gmt":"2013-10-11T01:28:02","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=47386"},"modified":"2021-10-06T16:20:08","modified_gmt":"2021-10-06T08:20:08","slug":"%e6%ad%a3%e5%bc%a6%e5%ae%9a%e5%be%8bthe-sine-law","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%ad%a3%e5%bc%a6%e5%ae%9a%e5%be%8bthe-sine-law\/","title":{"rendered":"\u6b63\u5f26\u5b9a\u5f8b (The Sine Law)"},"content":{"rendered":"

\u6b63\u5f26\u5b9a\u5f8b (The Sine Law)<\/strong><\/span>
\n\u81fa\u5317\u5e02\u7acb\u7b2c\u4e00\u5973\u5b50\u4e2d\u5b78\u6578\u5b78\u79d1\u8607\u4fca\u9d3b\u8001\u5e2b<\/strong><\/span><\/p>\n

\u5728\u73fe\u884c\u9ad8\u4e2d\u8ab2\u7a0b\u4e2d\uff0c\u5c0d\u65bc\u6b63\u5f26\u5b9a\u5f8b\u7684\u63a8\u5c0e\u5e38\u662f\u900f\u904e\u4e09\u89d2\u5f62\u9762\u7a4d\u516c\u5f0f\u70ba\u5a92\u4ecb\uff1a<\/p>\n

\u5982\u5716\u4e00\uff0c\u7d66\u5b9a\u4e09\u89d2\u5f62 \\(\\Delta ABC\\)\u00a0\uff0c\u5247\u4e09\u89d2\u5f62 \\(\\Delta ABC\\)\u00a0\u7684\u9762\u7a4d\u70ba<\/p>\n

\\(\\displaystyle\\frac{1}{2}ab\\sin C = \\frac{1}{2}ac\\sin B = \\frac{1}{2}bc\\sin A\\)<\/p>\n

\u56e0\u6b64\uff0c\\(\\displaystyle\\frac{{\\sin C}}{c} = \\frac{{\\sin B}}{b} = \\frac{{\\sin A}}{a} \\Rightarrow \\frac{a}{{\\sin A}} = \\frac{b}{{\\sin B}} = \\frac{c}{{\\sin C}}\\)<\/p>\n

\"47386_p1\"<\/a><\/p>\n


\n\u5373\u70ba\u6b63\u5f26\u5b9a\u5f8b\u3002\u4e5f\u5c31\u662f \\(a:b:c=\\sin A:\\sin B:\\sin C\\)\uff0c\u6eff\u8db3\u300c\u5927\u908a\u5c0d\u5927\u89d2\uff0c\u5c0f\u908a\u5c0d\u5c0f\u89d2\u300d\u5b9a\u91cf\u5316\u7684\u63cf\u8ff0\u3002\u5118\u7ba1\u9019\u500b\u63a8\u5c0e\u7684\u9032\u8def\u7c21\u55ae\u6613\u61c2\uff0c\u7136\u800c\uff0c\u8ac7\u5230\u6b63\u5f26\u5b9a\u5f8b\uff0c\u901a\u5e38\u9084\u6703\u63d0\u5230\u7b49\u5f0f\u7684\u6bd4\u503c\u70ba \\(2R\\)\uff0c\u5176\u4e2d \\(R\\)\u00a0\u70ba \\(\\Delta ABC\\)\u00a0\u7684\u5916\u63a5\u5713\u7684\u534a\u5f91\uff0c\u5373\u00a0\\(\\frac{a}{{\\sin A}} = \\frac{b}{{\\sin B}} = \\frac{c}{{\\sin C}} = 2R\\)\u3002<\/p>\n

\u4f46\u662f\uff0c\u9019\u500b\u6bd4\u503c\u5728\u4e0a\u8ff0\u7684\u63a8\u5c0e\u4e2d\u662f\u4e0d\u6703\u51fa\u73fe\u7684\u3002\u56e0\u6b64\uff0c\u82e5\u8981\u9069\u5207\u5730\u5f15\u5c0e\u51fa\u9019\u500b\u6027\u8cea\uff0c\u6050\u6015\u5f97\u8981\u8003\u616e\u5176\u4ed6\u7684\u6559\u5b78\u9032\u8def\u3002\u4e8b\u5be6\u4e0a\uff0c\u900f\u904e\u92b3\u89d2\u4e09\u89d2\u51fd\u6578\u7684\u5e7e\u4f55\u8868\u5fb5(\u8acb\u53c3\u95b1\u7db2\u7ad9\u53e6\u4e00\u7bc7\u6587\u7ae0\u3008\u92b3\u89d2\u4e09\u89d2\u51fd\u6578\u7684\u5e7e\u4f55\u8868\u5fb5\u3009)\uff0c\u662f\u53ef\u4ee5\u5713\u6eff\u5730\u89e3\u6c7a\u9019\u500b\u554f\u984c\u3002<\/p>\n

\"47386_p2\"<\/a>

\u5716\u4e8c<\/p><\/div>\n

\u7531\u65bc\u4e09\u89d2\u51fd\u6578\u53ef\u4ee5\u770b\u6210\u89d2\u5ea6\u5c0d\u61c9\u4e4b\u5713\u4e0a\u7684\u67d0\u4e9b\u7dda\u6bb5\u9577\uff0c\u4e0d\u59a8\u8003\u616e\u92b3\u89d2 \\(\\Delta ABC\\)\u00a0<\/em>\u53ca\u5176\u5916\u63a5\u5713(\u5982\u5716\u4e8c)\uff0c\u5247\u00a0\\(\\angle A = \\theta \\)\uff0c\\(\\displaystyle\\overline{CH}=\\frac{1}{2}a\\)\u3002\u56e0\u6b64\uff0c\u5728\u76f4\u89d2\u4e09\u89d2\u5f62 \\(\\Delta COH\\)\u00a0\u4e2d\uff0c<\/p>\n

\\(\\displaystyle\\frac{{\\frac{1}{2}a}}{R} = \\sin \\theta= \\sin A \\Rightarrow \\frac{a}{{\\sin A}} = 2R\\)<\/p>\n

\u540c\u7406\uff0c\\(\\displaystyle\\frac{b}{{\\sin B}} = \\frac{c}{{\\sin C}} = 2R\\)\u3002\u6240\u4ee5\uff0c\\(\\displaystyle\\frac{a}{{\\sin A}} = \\frac{b}{{\\sin B}} = \\frac{c}{{\\sin C}} = 2R\\)\u3002<\/p>\n

\"47386_p3\"<\/a>

\u5716\u4e09<\/p><\/div>\n

\u81f3\u65bc\u920d\u89d2\u4e09\u89d2\u5f62\u7684\u60c5\u6cc1\uff0c\u5247\u5982\u5716\u4e09\u3002\u5176\u4e2d\uff0c\u920d\u89d2\u00a0\\(\\angle A = \\frac{1}{2}(360^\\circ- 2\\theta ) = 180^\\circ- \\theta \\)<\/p>\n

\u56e0\u6b64\uff0c\\(\\displaystyle\\sin A = \\sin (180^\\circ- \\theta ) = \\sin \\theta= \\frac{{\\frac{1}{2}a}}{R} \\Rightarrow \\frac{a}{{\\sin A}} = 2R\\)\u3002<\/p>\n

\u82e5\u662f \\(\\angle B,~\\angle C\\)\uff0c\u53ea\u8981\u5ef6\u9577 \\(\\overline{OB}\\)\u00a0\u6216 \\(\\overline{OC}\\)\u00a0\u81f3\u5713\u4e0a\uff0c\u518d\u9023\u63a5 \\(A\\)\u00a0<\/i>\u5373\u5f97\u5716\u4e8c\u7684\u60c5\u5f62\u3002<\/p>\n

\u6700\u5f8c\uff0c\u5c0d\u65bc\u76f4\u89d2\u4e09\u89d2\u5f62\uff0c\u53ea\u662f\u7279\u4f8b\u60c5\u5f62\uff0c\u5c31\u8acb\u8b80\u8005\u81ea\u5df1\u52d5\u624b\u4e86\u3002\u5728\u7b46\u8005\u7684\u8ab2\u5802\u4e0a\uff0c\u5c07\u4e0a\u8ff0\u9032\u8def\u8a2d\u8a08\u6210\u5f15\u5c0e\u5f0f\u7684\u554f\u984c\uff0c\u5229\u7528\u5b78\u7fd2\u55ae\u8b93\u5b78\u751f\u5011\u5206\u7d44\u8a0e\u8ad6\uff0c\u901a\u5e38\u5b78\u751f\u90fd\u80fd\u9806\u5229\u63a8\u5c0e\u51fa\u6b63\u5f26\u5b9a\u5f8b\uff0c\u4eab\u53d7\u81ea\u5df1\u767c\u73fe\u5b9a\u7406\u7684\u6a02\u8da3\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u6b63\u5f26\u5b9a\u5f8b (The Sine Law) \u81fa\u5317\u5e02\u7acb\u7b2c\u4e00\u5973\u5b50\u4e2d\u5b78\u6578\u5b78\u79d1\u8607\u4fca\u9d3b\u8001\u5e2b \u5728\u73fe\u884c\u9ad8\u4e2d\u8ab2\u7a0b\u4e2d\uff0c\u5c0d\u65bc\u6b63\u5f26\u5b9a\u5f8b\u7684…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[222,219,111],"tags":[505,5549],"class_list":["post-47386","post","type-post","status-publish","format-standard","hentry","category-math03-03","category-math03","category-mathematics00","tag-505","tag-5549","loop-entry","cat-222","cat-219","cat-111","no-thumbnail"],"views":3771,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47386","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=47386"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47386\/revisions"}],"predecessor-version":[{"id":87456,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47386\/revisions\/87456"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=47386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=47386"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=47386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}